The number of diagonals in a polygon is $20$. The number of sides of the polygon is:

In how many ways can $5$ boys and $3$ girls sit in a row so that no two girls are together?

$p$ points are chosen on each of the three coplanar lines. The maximum number of triangles formed with vertices at these points is

There are $10$ points on the line segment $AB$ excluding $A$ and $B$,and $8$ points on the line segment $AC$ excluding $A$ and $C$. The number of triangles formed using these $18$ points (excluding $A, B,$ and $C$) is:

Out of $10$ points in a plane,$6$ are in a straight line. The number of triangles formed by joining these points is:

There are $6$ points on a circle. Two triangles are drawn such that they have no common vertices. What is the probability that none of the sides of the triangles intersect?